Universal central extensions of superdialgebras of matrices

arXiv:1602.04507v1 [math.RA] 14 Feb 2016

UNIVERSAL CENTRAL EXTENSIONS OF SUPERDIALGEBRAS OF MATRICES X. GARCÍA–MARTÍNEZ AND M. LADRA Abstract. We complete the problem of finding the universal central extension in the category of Leibniz superalgebras of sl(m, n, D) when m + n ≥ 3 and D is a superdialgebra, solving in particular the problem when D is an associative algebra, superalgebra or dialgebra. To accomplish this task we use a different method than the standard studied in the literature. We introduce and use the non-abelian tensor square of Leibniz superalgebras and its relations with the universal central extension.

1. Introduction Leibniz algebras, the non-antisymmetric analogue of Lie algebras, were first defined by Bloh [1] and later recovered by Loday in [22] when he handled periodicity phenomena in algebraic K-theory. Many authors have studied this structure and it has some interesting applications in Geometry and Physics ([16], [25], [6]). On the other hand, the theory of superalgebras arises directly from supersymmetry, a part of the theory of elemental particles, in order to have a better understanding of the geometrical structure of spacetime and to complete the substantial meaningful task of the unification of quantum theory and general relativity ([32]). The study of Lie or Leibniz superalgebras has been a very active field in the recent years since the classification of simple complex finite-dimensional Lie superalgebras by Kac in [14]. The study of central extensions is a very important topic in mathematics. There is a direct connection between central extensions and (co)homology, and they also have relations with Physics ([30]). In particular, universal central extensions have been studied in many different structures as groups [27], Lie algebras [11], [31] or Lie superalgebras [28]. A very interesting tool in the study of universal central extensions is the non-abelian tensor product introduced in [2] and extended to Lie algebras in [5] and to Lie superalgebras in [9]. The theory related with the universal central extension of the special linear algebra sl(n, A) has been very active due its relation with cyclic homology and its relevance in algebraic K-theory. The first approach was in the category of Lie algebras by Kassel and Loday in [15] where they described it when n ≥ 5 and A is an associative algebra, and in [8] it was obtained for n ≥ 3. For the Lie superalgebra 2010 Mathematics Subject Classification. 17B60, 17B55, 17B05. Key words and phrases. Leibniz (super)algebras, (Super)dialgebras, Universal central extensions. The authors were supported by Ministerio de Economía y Competitividad (Spain), grant MTM2013-43687-P (European FEDER support included) and by Xunta de Galicia, grant GRC2013-045 (European FEDER support included). The first author was also supported by FPU scholarship, Ministerio de Educación, Cultura y Deporte (Spain). 1

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sl(n, A) and A an associative superalgebra was given in [3]. For the special linear superalgebra sl(m, n, A) it was worked out for A an associative algebra in [26] and [29]; and for A an associative superalgebra in [4] and [10]. In the category of Leibniz algebras, the universal central extension of sl(n, A) (seen as a Leibniz algebra), where A is an associative algebra, was found in [24] when n ≥ 5 and in [13] when n ≥ 3. For the Leibniz superalgebra sl(m, n, A), when m + n ≥ 5 and A is an associative algebra it was calculated in [21]. In [18] it was found for the Leibniz algebra sl(m, D) and for the Leibniz superalgebra sl(m, n, D), where m ≥ 5 and m + n ≥ 5, respectively, and D is an associative dialgebra. The aim of this paper is to complete the task, finding the universal central extension of sl(m, n, D) where D is a superdialgebra and m + n ≥ 3. Since associative algebras, associative superalgebras and dialgebras are all examples of associative superdialgebras, we will solve all cases at once. Moreover, we obtain a result contradicting a specific point of a theorem given in [18]. The most interesting part of this paper is that the method used is not the same as in all the papers cited above. Due its relation with central extensions, we introduce and use the non abelian tensor square of Leibniz superalgebras providing another point of view to this topic. 2. Preliminaries In what follows we fix a unital commutative ring R. 2.1. Dialgebras. We recall from [23] the definitions and basic examples of (super)dialgebras. Definition 2.1. An associative dialgebra (dialgebra for short) is an R-module equipped with two R-linear maps ⊢ : D ⊗R D → D, ⊣ : D ⊗R D → D, where ⊢ and ⊣ are associative and satisfy the following conditions:   a ⊣ (b ⊣ c) = a ⊣ (b ⊢ c), (a ⊢ b) ⊣ c = a ⊢ (b ⊣ c),   (a ⊣ b) ⊢ c = (a ⊢ b) ⊢ c, for all a, b, c ∈ D. A bar-unit in D is an element e ∈ D such that for all x ∈ D, a ⊣ e = a = e ⊢ a. Note that a bar-unit may not be unique. A unital dialgebra is a dialgebra with a chosen bar-unit, that will be denoted by 1. An ideal I ⊂ D is an R-submodule such that if x or y belong to I then x ⊣ y ∈ D and x ⊢ y ∈ D. An associative superdialgebra (superdialgebra for short) is a dialgebra equipped with a Z2 -graded structure compatible with the two operations, i.e. Dα¯ ⊢ Dβ¯ ⊆ Dα+ ¯ , β¯ ∈ Z2 . The concepts of bar-unit, unital and ¯ ⊣ Dβ¯ ⊆ Dα+ ¯ β¯ and Dα ¯ β¯ , for α ideal are analogous in superdialgebras. Note that the bar-unit is always even. Example 2.2. An associative (super)algebra defines a (super)dialgebra structure in a canonical way, where a ⊣ b = ab = a ⊢ b. If it is unital, then the superdialgebra is unital.

UNIVERSAL CENTRAL EXTENSIONS OF SUPERDIALGEBRAS OF MATRICES

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Example 2.3. Let (A, d) a differential associative (super)algebra, i.e., d(ab) = d(a)b + ad(b) and d2 = 0. We define the two operations by x ⊣ y = xd(y) x ⊢ y = d(x)y. It is immediate to check that with these operations (A, d) is a (super)dialgebra. Example 2.4. Let A an associative (super)algebra, M an A-(super)bimodule and f : M → A an A-(super)bimodule map. Then we can define a (super)dialgebra structure with operations m ⊣ m′ = mf (m′ ), m ⊢ m′ = f (m)m′ . Example 2.5. Let D and D′ be two superdialgebras. Then the tensor product D ⊗R D′ is a superdialgebra where ′

(a ⊗ a′ ) ⊣ (b ⊗ b′ ) = (−1)|a ||b| (a ⊣ b) ⊗ (a′ ⊣ b′ ), ′

(a ⊗ a′ ) ⊢ (b ⊗ b′ ) = (−1)|a ||b| (a ⊢ b) ⊗ (a′ ⊢ b′ ). Example 2.6. A particular case of the previous example is M(n, D) = M(n, R)⊗R D, the R-supermodule of (n × n)-matrices. The operations are given by X X (a ⊣ b)ij = aik ⊣ bkj and (a ⊢ b)ij = aik ⊢ bkj . k

k

2.2. Leibniz superalgebras.

Definition 2.7. A Leibniz superalgebra L is an R-supermodule with an R-linear even map [−, −] : L ⊗R L → L, satisfying the Leibniz identity       x, [y, z] = [x, y], z − (−1)|y||z| [x, z], y , for all x, y, z ∈ L.

Note that a Leibniz superalgebra where the identity [x, y] = −(−1)|x||y|[y, x] also holds, is a Lie superalgebra. Example 2.8. A Lie superalgebra is in particular a Leibniz superalgebra. Example 2.9. Let D be a superdialgebra. Then D with the bracket [a, b] = a ⊣ b − (−1)|a||b| b ⊢ a, is a Leibniz superalgebra. If the two operations ⊣ and ⊢ are equal, i.e., D is also an associative superalgebra, this bracket also induces a Lie superalgebra structure. Definition 2.10. The centre of a Leibniz superalgebra L, denoted by Z(L), is the ideal formed by the elements z ∈ L such that [z, x] = [x, z] = 0 for all x ∈ L. The commutator of L, denoted by [L, L], is the ideal generated by the elements [x, y] where x, y ∈ L. A Leibniz superalgebra is called perfect if L = [L, L]. Definition 2.11. A central extension of a Leibniz superalgebra L is a surjective homomorphism φ : M → L such that Ker φ ⊆ Z(M ). We say that a central extension u : U → L is universal if for any central extension φ : M → L there is a unique homomorphism f : U → M such that u = φ ◦ f .

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X. GARCÍA–MARTÍNEZ AND M. LADRA

The theory of central extensions of Leibniz superalgebras is studied in [20]. We obtain the following straightforward results. Proposition 2.12. Let φ : E → M and ψ : M → L be two central extensions of Leibniz superalgebras. Then φ is universal if and only if ψ ◦ φ is universal. Proposition 2.13. Let M be a Leibniz superalgebra and L an R-supermodule. An R-supermodule homomorphism ϕ : M → L such that Ker ϕ ⊆ Z(M ) defines a Leibniz superalgebra structure in L where the bracket is [x, y] = ϕ([ϕ−1 (x), ϕ−1 (y)]). for x, y ∈ L. Now we introduce the homology of Leibniz superalgebras with trivial coefficients adapting it from the non-graded version [24]. Definition 2.14. Let L be a Leibniz superalgebra and δn : L⊗n → L⊗n−1 the R-linear map given by δn (x1 ⊗ · · · ⊗ xn ) = X (−1)n−j+|xj |(|xi+1 |+···+|xj−1 |) x1 ⊗· · ·⊗xi−1 ⊗[xi , xj ]⊗xi+1 ⊗· · ·⊗ x ˆj ⊗· · ·⊗xn . i